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I would like to announce the GAP3 file PrimitiveInvariant.g, which

contains programs to compute primitive invariants, polynomials that

characterize subgroups of the symmetric group and that are of use in

Galois theory. The main algorithms encoded by the programs generalize

methods of K. Girstmair for calculating minimal primitive invariants.

Let L and H be subgroups of the symmetric group of degree n, with H

contained in L. The function MinimalPrimitiveInvariants(n,L,H) computes

all representatives of H-invariant L-primitive polynomials of minimal

degree. The function AllPrimitiveInvariants(n,L,H) computes all

representatives of H-invariant L-primitive polynomials of degree up to

n(n-1)/2. In applications to resolvant calculation in Galois theory, n is

the degree of the polynomial, L the candidate group and H the test group.

The file is divided into two parts. The first part contains functions for

calculating with combinations of lists that represent indices of monomials

in n variables. These lists give a GAP3 way to handle polynomials, which

are dealt with differently in GAP4. The second part of the file contains

functions to compute what one might call the essential partitions and

sets, as well as the main functions.

PrimitiveInvariant.g is available in the "deposit" ftp subdirectory, as

pub/gap/gap-3.4.4/deposit/gap/priminv.g

or by following the links "Get GAP...more" and "GAP3 User contributions"

from the GAP main Web page, to arrive at the URL

www-history.mcs.st-and.ac.uk/~gap/Info/deposit.html.

The article "Calculs d'Invariants Primitifs de Groupes finis" which gives

the theory behind the algorithms will appear in RAIRO -Informatique

Theorique et Programmation and is also available at

www-calfor.lip6.fr/~abdeljao.

I hope that GAP users will find these programs useful. I would be glad to

correspond further with anybody who has questions about them or has

applications for them.

Ines Abdeljaouad

< Ines.Abdeljaouad@lip6.fr >

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